First Derivative Test

Suppose $f(x)$ is Continuous at a Stationary Point $x_0$.

1. If $f'(x) > 0$ on an Open Interval extending left from $x_0$ and $f'(x) < 0$ on an Open Interval extending right from $x_0$, then $f(x)$ has a Relative Maximum (possibly a Global Maximum) at $x_0$.

2. If $f'(x) < 0$ on an Open Interval extending left from $x_0$ and $f'(x) > 0$ on an Open Interval extending right from $x_0$, then $f(x)$ has a Relative Minimum (possibly a Global Minimum) at $x_0$.

3. If $f'(x)$ has the same sign on an Open Interval extending left from $x_0$ and on an Open Interval extending right from $x_0$, then $f(x)$ does not have a Relative Extremum at $x_0$.

See also Extremum, Global Maximum, Global Minimum, Inflection Point, Maximum, Minimum, Relative Extremum, Relative Maximum, Relative Minimum, Second Derivative Test, Stationary Point

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 14, 1972.